PHYS101: Introduction to Mechanics

Often in science, we deal with measurements that are very large or very small. When writing these numbers or doing calculations with these physical quantities, you would have to write a large number of zeros either at the end of a large value or at the beginning of a very small value. Scientific notation allows us to write these large or small numbers without writing all the "placeholder" zeros. We write the non-zero part of the value as a decimal, followed by an exponent showing the order of magnitude, or number of zeros before or after the number.

For example, consider the measurement: 125000 m. To write this measurement in scientific notation, we first take the non-zero part of the number, and write it as a decimal. The decimal part of the number above would become 1.25. Then, we need to show the order of magnitude of the number. We count the number of decimal places from where we placed the decimal to the end of the number. In this case, there are five places between the decimal we put in and the end of the number. We write this as an exponent: $10^{5}$. To put the entire scientific notation together, we write: $1.25 \times 10^{5}\ \mathrm{m}$.

We can also do an example where the measurement is very small. For example, consider the measurement: 0.0000085 s. Here, we again begin by making the non-zero part of the number into a decimal. We would write: 8.5. Next, we need to show the order of magnitude of the number. For a small number (less than one), we count the number of places from where we wrote the decimal back to the original decimal place. Then, we write our exponent as a negative number to show that the number is less than one. For this example, the exponent is: $10^{-6}$. To put the entire scientific notation together, we write: $8.5 \times 10^{-6}\ \mathrm{m}$.

We can also convert values written in scientific notation to decimal notation. Consider the number: $5.0 \times 10^{3}\ \mathrm{m}$. We can write this as normal notation by adding the appropriate number of decimal places to the number, past the decimal written in scientific notation. Here, the order of magnitude (number of decimal places) is three, as we see from the exponent part of the number. Because the exponent is positive, we add the decimal places to the right of the number to make it a large number. The value in normal notation is: $5000\ \mathrm{m}$.

We can also do this for small numbers written in scientific notation. Consider the example: $4.2 \times 10^{-4}\ \mathrm{m}$. We can write this as normal notation by adding the appropriate number of decimal places to the left of the number to make it a small number. Here, we need to have four decimal places to the left of the decimal in the scientific notation. The value in normal notation is: $0.00042\ \mathrm{m}$.